From 79544c81eb5525ebc0e5a930c897d6d2e141481e Mon Sep 17 00:00:00 2001 From: Saahil Mahato <115351000+saahil-mahato@users.noreply.github.com> Date: Fri, 11 Oct 2024 02:26:58 +0545 Subject: [PATCH] feat: add solovay strassen primality test (#5692) * feat: add solovay strassen primality test * chore: add wikipedia link * fix: format and coverage * fix: mvn stylecheck * fix: PMD errors * refactor: make random final --------- Co-authored-by: Alex Klymenko --- .../maths/SolovayStrassenPrimalityTest.java | 133 ++++++++++++++++++ .../SolovayStrassenPrimalityTestTest.java | 122 ++++++++++++++++ 2 files changed, 255 insertions(+) create mode 100644 src/main/java/com/thealgorithms/maths/SolovayStrassenPrimalityTest.java create mode 100644 src/test/java/com/thealgorithms/maths/SolovayStrassenPrimalityTestTest.java diff --git a/src/main/java/com/thealgorithms/maths/SolovayStrassenPrimalityTest.java b/src/main/java/com/thealgorithms/maths/SolovayStrassenPrimalityTest.java new file mode 100644 index 000000000..caa1abfc3 --- /dev/null +++ b/src/main/java/com/thealgorithms/maths/SolovayStrassenPrimalityTest.java @@ -0,0 +1,133 @@ +package com.thealgorithms.maths; + +import java.util.Random; + +/** + * This class implements the Solovay-Strassen primality test, + * which is a probabilistic algorithm to determine whether a number is prime. + * The algorithm is based on properties of the Jacobi symbol and modular exponentiation. + * + * For more information, go to {@link https://en.wikipedia.org/wiki/Solovay%E2%80%93Strassen_primality_test} + */ +final class SolovayStrassenPrimalityTest { + + private final Random random; + + /** + * Constructs a SolovayStrassenPrimalityTest instance with a specified seed for randomness. + * + * @param seed the seed for generating random numbers + */ + private SolovayStrassenPrimalityTest(int seed) { + random = new Random(seed); + } + + /** + * Factory method to create an instance of SolovayStrassenPrimalityTest. + * + * @param seed the seed for generating random numbers + * @return a new instance of SolovayStrassenPrimalityTest + */ + public static SolovayStrassenPrimalityTest getSolovayStrassenPrimalityTest(int seed) { + return new SolovayStrassenPrimalityTest(seed); + } + + /** + * Calculates modular exponentiation using the method of exponentiation by squaring. + * + * @param base the base number + * @param exponent the exponent + * @param mod the modulus + * @return (base^exponent) mod mod + */ + private static long calculateModularExponentiation(long base, long exponent, long mod) { + long x = 1; // This will hold the result of (base^exponent) % mod + long y = base; // This holds the current base value being squared + + while (exponent > 0) { + // If exponent is odd, multiply the current base (y) with x + if (exponent % 2 == 1) { + x = x * y % mod; // Update result with current base + } + // Square the base for the next iteration + y = y * y % mod; // Update base to be y^2 + exponent = exponent / 2; // Halve the exponent for next iteration + } + + return x % mod; // Return final result after all iterations + } + + /** + * Computes the Jacobi symbol (a/n), which is a generalization of the Legendre symbol. + * + * @param a the numerator + * @param num the denominator (must be an odd positive integer) + * @return the Jacobi symbol value: 1, -1, or 0 + */ + public int calculateJacobi(long a, long num) { + // Check if num is non-positive or even; Jacobi symbol is not defined in these cases + if (num <= 0 || num % 2 == 0) { + return 0; + } + + a = a % num; // Reduce a modulo num to simplify calculations + int jacobi = 1; // Initialize Jacobi symbol value + + while (a != 0) { + // While a is even, reduce it and adjust jacobi based on properties of num + while (a % 2 == 0) { + a /= 2; // Divide a by 2 until it becomes odd + long nMod8 = num % 8; // Get num modulo 8 to check conditions for jacobi adjustment + if (nMod8 == 3 || nMod8 == 5) { + jacobi = -jacobi; // Flip jacobi sign based on properties of num modulo 8 + } + } + + long temp = a; // Temporarily store value of a + a = num; // Set a to be num for next iteration + num = temp; // Set num to be previous value of a + + // Adjust jacobi based on properties of both numbers when both are odd and congruent to 3 modulo 4 + if (a % 4 == 3 && num % 4 == 3) { + jacobi = -jacobi; // Flip jacobi sign again based on congruences + } + + a = a % num; // Reduce a modulo num for next iteration of Jacobi computation + } + + return (num == 1) ? jacobi : 0; // If num reduces to 1, return jacobi value, otherwise return 0 (not defined) + } + + /** + * Performs the Solovay-Strassen primality test on a given number. + * + * @param num the number to be tested for primality + * @param iterations the number of iterations to run for accuracy + * @return true if num is likely prime, false if it is composite + */ + public boolean solovayStrassen(long num, int iterations) { + if (num <= 1) { + return false; // Numbers <=1 are not prime by definition. + } + if (num <= 3) { + return true; // Numbers <=3 are prime. + } + + for (int i = 0; i < iterations; i++) { + long r = Math.abs(random.nextLong() % (num - 1)) + 2; // Generate a non-negative random number. + long a = r % (num - 1) + 1; // Choose random 'a' in range [1, n-1]. + + long jacobi = (num + calculateJacobi(a, num)) % num; + // Calculate Jacobi symbol and adjust it modulo n. + + long mod = calculateModularExponentiation(a, (num - 1) / 2, num); + // Calculate modular exponentiation: a^((n-1)/2) mod n. + + if (jacobi == 0 || mod != jacobi) { + return false; // If Jacobi symbol is zero or doesn't match modular result, n is composite. + } + } + + return true; // If no contradictions found after all iterations, n is likely prime. + } +} diff --git a/src/test/java/com/thealgorithms/maths/SolovayStrassenPrimalityTestTest.java b/src/test/java/com/thealgorithms/maths/SolovayStrassenPrimalityTestTest.java new file mode 100644 index 000000000..18cc35266 --- /dev/null +++ b/src/test/java/com/thealgorithms/maths/SolovayStrassenPrimalityTestTest.java @@ -0,0 +1,122 @@ +package com.thealgorithms.maths; + +import static org.junit.jupiter.api.Assertions.assertEquals; +import static org.junit.jupiter.api.Assertions.assertFalse; +import static org.junit.jupiter.api.Assertions.assertTrue; + +import org.junit.jupiter.api.BeforeEach; +import org.junit.jupiter.api.Test; +import org.junit.jupiter.params.ParameterizedTest; +import org.junit.jupiter.params.provider.MethodSource; + +/** + * Unit tests for the {@link SolovayStrassenPrimalityTest} class. + * This class tests the functionality of the Solovay-Strassen primality test implementation. + */ +class SolovayStrassenPrimalityTestTest { + + private static final int RANDOM_SEED = 123; // Seed for reproducibility + private SolovayStrassenPrimalityTest testInstance; + + /** + * Sets up a new instance of {@link SolovayStrassenPrimalityTest} + * before each test case, using a fixed random seed for consistency. + */ + @BeforeEach + void setUp() { + testInstance = SolovayStrassenPrimalityTest.getSolovayStrassenPrimalityTest(RANDOM_SEED); + } + + /** + * Provides test cases for prime numbers with various values of n and k (iterations). + * + * @return an array of objects containing pairs of n and k values + */ + static Object[][] primeNumbers() { + return new Object[][] {{2, 1}, {3, 1}, {5, 5}, {7, 10}, {11, 20}, {13, 10}, {17, 5}, {19, 1}}; + } + + /** + * Tests known prime numbers with various values of n and k (iterations). + * + * @param n the number to be tested for primality + * @param k the number of iterations to use in the primality test + */ + @ParameterizedTest + @MethodSource("primeNumbers") + void testPrimeNumbersWithDifferentNAndK(int n, int k) { + assertTrue(testInstance.solovayStrassen(n, k), n + " should be prime"); + } + + /** + * Provides test cases for composite numbers with various values of n and k (iterations). + * + * @return an array of objects containing pairs of n and k values + */ + static Object[][] compositeNumbers() { + return new Object[][] {{4, 1}, {6, 5}, {8, 10}, {9, 20}, {10, 1}, {12, 5}, {15, 10}}; + } + + /** + * Tests known composite numbers with various values of n and k (iterations). + * + * @param n the number to be tested for primality + * @param k the number of iterations to use in the primality test + */ + @ParameterizedTest + @MethodSource("compositeNumbers") + void testCompositeNumbersWithDifferentNAndK(int n, int k) { + assertFalse(testInstance.solovayStrassen(n, k), n + " should be composite"); + } + + /** + * Tests edge cases for the primality test. + * This includes negative numbers and small integers (0 and 1). + */ + @Test + void testEdgeCases() { + assertFalse(testInstance.solovayStrassen(-1, 10), "-1 should not be prime"); + assertFalse(testInstance.solovayStrassen(0, 10), "0 should not be prime"); + assertFalse(testInstance.solovayStrassen(1, 10), "1 should not be prime"); + + // Test small primes and composites + assertTrue(testInstance.solovayStrassen(2, 1), "2 is a prime number (single iteration)"); + assertFalse(testInstance.solovayStrassen(9, 1), "9 is a composite number (single iteration)"); + + // Test larger primes and composites + long largePrime = 104729; // Known large prime number + long largeComposite = 104730; // Composite number (even) + + assertTrue(testInstance.solovayStrassen(largePrime, 20), "104729 is a prime number"); + assertFalse(testInstance.solovayStrassen(largeComposite, 20), "104730 is a composite number"); + + // Test very large numbers (may take longer) + long veryLargePrime = 512927357; // Known very large prime number + long veryLargeComposite = 512927358; // Composite number (even) + + assertTrue(testInstance.solovayStrassen(veryLargePrime, 20), Long.MAX_VALUE - 1 + " is likely a prime number."); + + assertFalse(testInstance.solovayStrassen(veryLargeComposite, 20), Long.MAX_VALUE + " is a composite number."); + } + + /** + * Tests the Jacobi symbol calculation directly for known values. + * This verifies that the Jacobi symbol method behaves as expected. + */ + @Test + void testJacobiSymbolCalculation() { + // Jacobi symbol (a/n) where n is odd and positive + int jacobi1 = testInstance.calculateJacobi(6, 11); // Should return -1 + int jacobi2 = testInstance.calculateJacobi(5, 11); // Should return +1 + + assertEquals(-1, jacobi1); + assertEquals(+1, jacobi2); + + // Edge case: Jacobi symbol with even n or non-positive n + int jacobi4 = testInstance.calculateJacobi(5, -11); // Should return 0 (invalid) + int jacobi5 = testInstance.calculateJacobi(5, 0); // Should return 0 (invalid) + + assertEquals(0, jacobi4); + assertEquals(0, jacobi5); + } +}